This is why I think a birth certificate should reflect the biological truth as transparently as is possible.
Which means the most key information is 1. The baby’s name and 2. The mother - ‘mater’, ie, the woman’s whose womb - ‘matrix’ the baby was born from, 3. The date and 4. The place of birth.
Not essential, but the biological father (if present at the registration and/or married to the mother) is normally included. PPs have eloquently explained why they should not always be included.
And in a situation where donor gametes are used, the genetic mother or father should be registered as ‘additional information’ for factual accuracy, without conferring any parental rights upon them. This extra information is only necessary because modern interventional methods of conception, using donors, has skewed the definition of what mother and father has always meant, and it is only fair that the birth certificate should equally reflect all the information a non-interventional conception would show (which includes genetic parentage) and not obscure the truth.
People who are not biologically related do not belong on the birth certificate. Just because historically some men might claim to be a father when they aren’t (to make an ‘honest woman’ of the mother) or because the mother may have been unfaithful and doesn’t want to disclose it, those who partake in this probably unlawful deception don’t change the intended meaning of the named father on the birth certificate to now be ‘anyone who shows up to claim paternity or is married to the mother’. The intention is that it should be truthful and accurate, that the father really is the father - biologically.
People who are not biologically related don’t belong on a birth certificate and they should apply to adopt as a step parent. That way it is clear and transparent. They are not a biological parent, but they do have legal parental responsibility.
If the truth about parentage is fudged on the birth certificate for the convenience, control and sentimentality of a couple in one situation, surely should be the same for everyone? That way lies surrogacy, GRA, genetic bewilderment, etc, etc.